Math 3626 Midterm Exam – Review Topics
You should be familiar with the following concepts:
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Matrix theory (matrix algebra, determinant, Eigenvalues, Eigenvectors, inverse, etc)
Solving linear systems of equations
Rotating 2D and 3D vectors; Rotational matrices
Linear Programming and the Simplex method: Setting up initial tableau, find pivot element, pivot
operation, reading off final solution, the “dual” problem
Complexity of algorithms
Recursive Algorithms
1st order ODE and separable DE
Higher-order DE’s and converting them to systems of linear Des
Homogeneous systems of 1st order linear DE’s: Solution in 2D in case of two real Eigenvalues and 2
complex ones; 3 x 3 systems with three different real Eigenvalues; solve initial value problems.
Sample Problems:
1
1. Consider the matrix 𝐴 = (
0
Mathematica)
2
). Find (a) determinant, (b) Eigenvalues, (c) Eigenvectors, (d) inverse (no
3
1 0
2. Same with the 3 x 3 matrix 𝐴 = ( 0 1
−3 2
−3
2 ) (with or without Mathematica)
1
3. Solve the following system of equations. If there is no unique solution, state if the system has infinitely
many or no solution:
𝑥+𝑦 =1
𝑥 + 𝑧 = −1
𝑦+𝑤 =1
2𝑥 + 𝑦 + 𝑧 = 1
4. Solve the following system of equations (with or without Mathematica)
𝑥 + 2𝑦 = 4
3𝑦 = 6
𝑥 − 3𝑧 = 6
𝑦 + 2𝑧 = 10
−3𝑥 + 2𝑦 + 𝑧 = 18
5. Find the initial tableau for the optimization problem to maximize 2x + 4y + z subject to the constraints x
+ 3y <= 10, 4x + y + 3z <= 8, x + 6z <= 12 and x,y,z >= 0. Note: you do not need to solve the problem, just
set it up properly. Then circle the first pivot element. Finally, perform one complete pivot operation.
6. Below are two matrices utilizing the simplex method, both using three variables x, y, and z. Which one is
done, and what is the answer represented by that matrix? For the matrix that’s not done, circle the next
pivot element (but you don’t need to pivot):
7
4
3
−4
9
2
29
( 4
0 1
1 0
0 0
0 0
1
4
1
−4
1
2
7
4
3
4
1
4
1
2
9
4
0 0
0 0
1 0
0 1
33
4
7
4
33
2
111
4 )
3
10
61
10
43
− 10
(
9
10
73
10
11
10
1
0
0
1
10
3
− 10
9
10
0 0
1
1 0
18
0 1
9
)
7. Solve the following optimization problems:
a. Maximize 4 x + 2 y subject to 10 x + 5 y <= 50, 12 x + y <= 48, 3 x + 2 y <= 18, x >= 0, and y >= 0.
b. Maximize 2 x - 3 y + 4 z subject to 4 x - 3 y + z <= 3, x + y + z <= 10, 2 x + y - z <= 10, x >= 0, y >= 0, and
z >= 0
8. Solve the problem: minimize x + 3 y but make sure that x + y >= 1 and 4x + 2y >= 3, and of course x >= 0
and y >= 0. Hint: check your notes or the available online resources for the “dual problem”.
9. In class we computed the rotational matrix Rx(x) to rotate around the x-axis by x degrees, Ry(y) to rotate
around the y axis by y degrees, and Rz(z) around the z-axis by z degrees. We came up with R(x,y,z) to be:
List the vector <1,2,3> after rotating it by (a) Pi/4 around x, and then by (b) pi/3 about z-axis
10. Consider the following functions. List their order of complexity, i.e. find simple functions g such that f =
O(g):
a. 𝑓(𝑛) = 3𝑛4 + 2𝑛2 − 10
b. 𝑓(𝑛) = 𝑛2 + 2𝑛 + log 2 (𝑛)
3
c. 𝑓(𝑛) = √𝑛 + √𝑛2 + ln⁡(𝑛)
11. What is the order of complexity of (a) adding two matrices, (b) multiplying two matrices, (c) computing
the determinant of a matrix using our “standard” algorithm?
12. Solve the following separable 1st order DE: 𝑦 ′ = 6𝑦 2 𝑥 with initial condition 𝑦(6) = −
′
3
1
.
80
What happens
to the solution y as x goes to infinity? Do the same for 𝑦 √1 + 𝑥 2 = 𝑥𝑦 with 𝑦(0) = −1
13. Convert the following higher order DE into systems of 1st order DE
𝑦 ′′′ − 5𝑦 ′′ + 9𝑦 = 𝑡 cos(2𝑡)
𝑦 (4) + 3𝑦 ′′′ + 𝜋𝑦 ′ − 6𝑦 = 11
Is your ODE system conversion unique?
14. Solve the following system of ODE’s:
𝑥 ′ = 2𝑥 + 𝑦3
𝑦′ = 4𝑥 + 3𝑦
𝑋(0) =< 10,20 >
𝑥 ′ = −𝑥 − 6𝑦
𝑦 ′ = 3𝑥 + 5𝑦
1 −2 0
𝑋 ′ = 𝐴⁡𝑋⁡𝑤ℎ𝑒𝑟𝑒⁡𝐴 = (−2 2 5)
0
5 1
15. Solve the following 2nd Order DE: 𝑦 ′′ + 2𝑦 ′ = 7𝑦 and find all solutions with initial condition y(0) = 1
Mathematica Functions
You should be ready to explain the following Mathematica functions and use any of them, as required, for
particular programming problems.
Defining a Function
:=
List (Matrix) related
Length
AppendTo
Join
Drop
[[ ]], [[n]], [[All, n]], [[n,m]]. [[n;m]]
Max
Min
Position
Map
Table
IdentityMatrix
RandomInteger
Eigenvalues
Eigenvectors
Graphics related
Graphics
Graphics3D
Polygon
Line
RGBColor
Plot
Plot3D
ListAnimate
RegionPlot
Programming
Do
While
If
Print
Timing
Logical Operators
==
!=
<, >
<=, >=
&&
||
Misc
Inverse
Det
Minimize
Maximize
DSolve